\(\int \frac {1}{x (c+a^2 c x^2)^{3/2} \arctan (a x)} \, dx\) [512]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\text {Int}\left (\frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)},x\right ) \]

[Out]

Unintegrable(1/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x)

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx \]

[In]

Int[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]

[Out]

Defer[Int][1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx \]

[In]

Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]),x]

[Out]

Integrate[1/(x*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]), x]

Maple [N/A] (verified)

Not integrable

Time = 2.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \arctan \left (a x \right )}d x\]

[In]

int(1/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x)

[Out]

int(1/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x \arctan \left (a x\right )} \,d x } \]

[In]

integrate(1/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x, algorithm="fricas")

[Out]

integral(sqrt(a^2*c*x^2 + c)/((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*arctan(a*x)), x)

Sympy [N/A]

Not integrable

Time = 3.83 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int \frac {1}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}}\, dx \]

[In]

integrate(1/x/(a**2*c*x**2+c)**(3/2)/atan(a*x),x)

[Out]

Integral(1/(x*(c*(a**2*x**2 + 1))**(3/2)*atan(a*x)), x)

Maxima [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x \arctan \left (a x\right )} \,d x } \]

[In]

integrate(1/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x, algorithm="maxima")

[Out]

integrate(1/((a^2*c*x^2 + c)^(3/2)*x*arctan(a*x)), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/x/(a^2*c*x^2+c)^(3/2)/arctan(a*x),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx=\int \frac {1}{x\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

int(1/(x*atan(a*x)*(c + a^2*c*x^2)^(3/2)),x)

[Out]

int(1/(x*atan(a*x)*(c + a^2*c*x^2)^(3/2)), x)